ON THE KOPLIENKO SPECTRAL SHIFT
FUNCTION, I. BASICS
FRITZ GESZTESY1 , ALEXANDER PUSHNITSKI2 , AND BARRY SIMON3
Abstract. We study the Koplienko Spectral Shift Function
(KoSSF), which is distinct from the one of Krein (KrSSF). KoSSF
is defined for pairs A, B with (A − B) ∈ I2 , the Hilbert–Schmidt
operators, while KrSSF is defined for pairs A, B with (A−B) ∈ I1 ,
the trace class operators. We review various aspects of the construction of both KoSSF and KrSSF. Among our new results are:
(i) that any positive Riemann integrable function of compact support occurs as a KoSSF; (ii) that there exist A, B with (A−B) ∈ I2
so det2 ((A − z)(B − z)−1 ) does not have nontangential boundary
values; (iii) an alternative definition of KoSSF in the unitary case;
and (iv) a new proof of the invariance of the a.c. spectrum under
I1 -perturbations that uses the KrSSF.
1. Introduction
In 1941, Titchmarsh [63] (see also [20, pp. 1564–1566] for the result)
proved that if
V ∈ L1 ((0, ∞); dx), V real-valued,
Date: May 16, 2007.
2000 Mathematics Subject Classification. Primary: 47A10, 81Q10. Secondary:
34B27, 47A40, 81Uxx.
Key words and phrases. Krein’s spectral shift function, Koplienko’s spectral shift
function, self-adjoint operators, trace class and Hilbert–Schmidt perturbations, convexity properties, boundary values of (modified) Fredholm determinants.
1
Department of Mathematics, University of Missouri, Columbia, MO 65211,
USA. E-mail: fritz@math.mizzou.edu. Supported in part by NSF Grant DMS0405526.
2
Department of Mathematics, King’s College London, Strand, London WC2R
2LS, England, UK. E-mail: alexander.pushnitski@kcl.ac.uk. Supported in part by
the Leverhulme Trust.
3
Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125,
USA. E-mail: bsimon@caltech.edu. Supported in part by NSF Grant DMS-0140592
and U.S.–Israel Binational Science Foundation (BSF) Grant No. 2002068.
Submitted to the Marchenko and Pastur birthday issue of Journal of Mathematical Physics, Analysis and Geometry.
1
2
F. GESZTESY, A. PUSHNITSKI, AND B. SIMON
and
d2
+ V,
(1.1)
dx2
dom(Hθ ) = {f ∈ L2 ((0, ∞); dx) | f, f ′ ∈ AC([0, R]) for all R > 0;
Hθ = −
sin(θ)f ′ (0) + cos(θ)f (0) = 0; (−f ′′ + V f ) ∈ L2 ((0, ∞); dx)},
for some θ ∈ [0, π), then
σac (Hθ ) = [0, ∞).
(Actually, he explicitly computed the spectral function in terms of the
inverse square of the modulus of the Jost function for positive energies.)
It was later realized that the a.c. invariance, that is,
σac (Hθ ) = σac (H0,θ )
(1.2)
with
d2
,
(1.3)
dx2
dom(H0,θ ) = {f ∈ L2 ((0, ∞); dx) | f, f ′ ∈ AC([0, R]) for all R > 0;
H0,θ = −
sin(θ)f ′ (0) + cos(θ)f (0) = 0; f ′′ ∈ L2 ((0, ∞); dx)},
is a special case of an invariance of the absolutely continuous spectrum,
σac (·) for the passage from A to B if (A − B) ∈ I1 , the trace class. In
the present context of the pair (Hθ , H0,θ ) one has [(Hθ + E)−1 − (H0,θ +
E)−1 ] ∈ I1 for E > 0 sufficiently large. The abstract trace class result
is associated with Birman [10, 11], Kato [31, 32], and Rosenblum [54].
Our original and continuing motivation is to find a suitable operator
theoretic result connected with the remarkable discovery of Deift–Killip
[18] that for the above (1.1)/(1.3) case, one has (1.2) if one only assumes
V ∈ L2 ((0, ∞); dx). Note that V ∈ L2 ((0, ∞); dx) implies that
[(Hθ + E)−1 − (H0,θ + E)−1 ] ∈ I2 ,
the Hilbert–Schmidt class. However, there is no totally general invariance result for a.c. spectrum under non-trace class perturbations: It is
a result of Weyl [67] and von Neumann [65] that given any self-adjoint
A, there is a B with pure point spectrum and (A − B) ∈ I2 . Kuroda
[43] extends this to Ip , p ∈ (1, ∞), the trace ideals. Thus, we seek
general operator criteria on when (A − B) ∈ I2 but (1.2) still holds.
We hope such a criterion will be found in the spectral shift function
of Koplienko [36] (henceforth KoSSF), an object which we believe has
not received the attention it deserves. One of our goals in the present
paper is to make propaganda for this object.
ON THE KOPLIENKO SPECTRAL SHIFT FUNCTION, I. BASICS
3
Two references for trace ideals we quote extensively are Gohberg–
Krein [27] and Simon [60]. We follow the notation of [60]. Throughout
this paper all Hilbert spaces are assumed to be complex and separable.
The KoSSF, η(λ; A, B), is defined when A and B are bounded selfadjoint operators satisfying (A − B) ∈ I2 , and is given by
Z
d
′′
f (λ)η(λ; A, B) dλ = Tr f (A) − f (B) −
f (B + α(A − B))
,
dα
R
α=0
(1.4)
where the right-hand side is sometimes (certainly if (A − B) ∈ I1 ) the
simpler-looking
Tr(f (A) − f (B) − (A − B)f ′ (B)).
(1.5)
η has two critical properties: η ∈ L1 (R) and η ≥ 0. We mainly consider
bounded A, B here, but see the remarks in Section 9.
Formula (1.4) requires some assumptions on f . In Koplienko’s original paper [36] the case f (x) = (x − z)−1 was considered and then
(1.4) was extended to the class of rational functions with poles off the
real axis. Later, Peller [52] extended the class of functions f and found
sharp sufficient conditions on f which guarantee that (1.4) holds. These
conditions were stated in terms of Besov spaces. Essentially, Peller’s
construction requires that (1.4) hold for some sufficiently wide class of
functions, so that this class is dense in a certain Besov space, and then
provides an extension onto the whole of this Besov space.
We will use this aspect of Peller’s work and will not worry about
the classes of f in this paper. For the most part we will work with
f ∈ C ∞ (R) and Peller’s construction provides an extension to a wider
function class.
The model for the KoSSF is, of course, the spectral shift function of
Krein (henceforth KrSSF), denoted by ξ(λ; A, B), and defined for A, B
with (A − B) ∈ I1 by
Z
ξ(λ; A, B)f ′(λ) dλ = Tr(f (A) − f (B)).
(1.6)
R
In the appendix, we recall a quick way to define ξ, its main properties
and, most importantly, present an argument that shows how it can
be used to derive the invariance of a.c. spectrum without recourse to
scattering theory.
As we will see in Section 2, it is easy to construct analogs of η for
any In , n ∈ N, but they are only tempered distributions. What makes
η different is its positivity, which also implies it lies in L1 (R) (by taking
f suitably). This positivity should be thought of as a general convexity
result—something hidden in Koplienko’s paper [36].
4
F. GESZTESY, A. PUSHNITSKI, AND B. SIMON
One of our goals here is to emphasize this convexity. Another is
to present a “baby” finite-dimensional version of the double Stieltjes
operator integral of Birman–Solomyak [12, 13, 15], essentially due to
Löwner [46], whose contribution here seems to have been overlooked.
In Section 2, we define η when (A−B) is trace class, and in Section 3,
we discuss the convexity result that is equivalent to positivity of η. In
Section 4, we prove a lovely bound of Birman–Solomyak [13, 15]:
kf (A) − f (B)kI2 ≤ kf ′ k∞ kA − BkI2 .
(1.7)
Here and in the remainder of this paper k · kIp denotes the norm in the
trace ideals Ip , p ∈ [1, ∞). In Section 5, we use (1.7) plus positivity of
η to complete the construction of η.
We want to emphasize an important distinction between the KrSSF
and the KoSSF. The former satisfies a chain rule
ξ( · ; A, C) = ξ( · ; A, B) + ξ( · ; B, C),
(1.8)
while η instead satisfies a corrected chain rule
η( · ; A, C) = η( · ; A, B) + η( · ; B, C) + δη( · ; A, B, C),
where δη satisfies
Z
g ′ (λ)δη(λ; A, B) dλ = Tr((A − B)(g(B) − g(C))).
(1.9)
(1.10)
R
(Here g corresponds to f ′ when comparing with (1.4)–(1.6).) It is in
estimating (1.10) that (1.7) will be critical.
We view Sections 2–5 as a repackaging in a prettier ribbon of Koplienko’s construction in [36]. Section 6 explores what η’s can occur. In
Sections 7 and 8, we discuss the connection to det2 (·) and present a new
result: an example of (A − B) ∈ I2 where det2 ((A − z)(B − z)−1 ) does
not have nontangential limits to the real axis a.e. This is in contradistinction to the KrSSF, where (A−B) ∈ I1 implies det((A−z)(B−z)−1 )
has a nontangential limit z → λ for a.e. λ ∈ R. The latter is a consequence of the formula
Z
−1
log(det((A − z)(B − z) )) = (λ − z)−1 ξ(λ; A, B) dλ, z ∈ C\R,
R
(1.11)
since the right-hand side of (1.11) represents a difference of two Herglotz functions.
Sections 9 and 10 discuss extensions of η to the case of unbounded
operators with a trace class condition on the resolvents and to unitary
operators. Here a key is that η is not determined until one makes a
choice of interpolation. Section 11 discusses some conjectures.
ON THE KOPLIENKO SPECTRAL SHIFT FUNCTION, I. BASICS
5
In a future joint work, we will explore what one can learn about
the KoSSF from Szegő’s theorem [59], the work of Killip–Simon [34]
and of Christ–Kiselev [17]. This will involve the study of η for suitable
Schrödinger operators and Jacobi and CMV matrices for perturbations
in Lp , respectively, ℓp , p ∈ [1, 2).
We are indebted to E. Lieb, K. A. Makarov, V. V. Peller, and
M. B. Ruskai for useful discussions. F. G. and A. P. wish to thank Gary
Lorden and Tom Tombrello for the hospitality of Caltech where some
of this work was done. F. G. gratefully acknowledges a research leave
for the academic year 2005/06 granted by the Research Council and
the Office of Research of the University of Missouri-Columbia. A. P.
gratefully acknowledges financial support by the Leverhulme Trust.
It is a great pleasure to dedicate this paper to the birthdays of two
giants of spectral theory: Vladimir A. Marchenko and Leonid A. Pastur.
2. The KoSSF η( · ; A, B) in the Trace Class Case
We begin with what can be said of In perturbations, n ∈ N, and
then turn to what is special for n = 1, 2. We note that our approach
has common elements to the one used by Dostanić [19].
Proposition 2.1. Let A, B be bounded self-adjoint operators with
A = B + X.
(2.1)
For α, t ∈ R, define
ft (α) = eit(B+αX) .
in α and
Then ft (α) is C ∞
Z
dk ft
k
=i
fs1 (α)Xfs2 (α) . . . fsk (α)Xft−s1 −···−sk (α) ds1 . . . dsk .
0<s <t
dαk
Pk j
j=1 sj <t
(2.2)
k
k
If X ∈ Ip for p ≥ k, k ∈ N, then d ft /dα ∈ Ip/k and
k d ft tk
≤
kXkkIp .
dαk k!
Ip/k
In particular, if n ∈ N and X ∈ In , then
k
n−1
X
1 d
itA
itB
gt (A, B) ≡ e − e −
ft (α)
∈ I1 ,
k!
dα
α=0
k=1
kgt (A, B)kI1 ≤
tn
kXknIn .
n!
(2.3)
(2.4)
(2.5)
6
F. GESZTESY, A. PUSHNITSKI, AND B. SIMON
Proof. For k = 1, (2.2) comes from taking limits in DuHamel’s formula
Z 1
C
D
e −e =
eβC (C − D)e(1−β)D dβ.
0
The general k case then follows by induction.
(2.2) implies (2.3) by Hölder’s inequality for operators (see [60, p.
21]). (2.4) is then Taylor’s theorem with remainder and (2.5) follows
from (2.3).
Theorem 2.2. Let A, B be bounded self-adjoint operators such that
X = (A − B) ∈ In for some n ∈ N. Let f be of compact support with
b its Fourier transform, satisfying
f,
Z
b
(1 + |k|)n |f(k)|
dk < ∞.
(2.6)
R
Then,
k
n−1
X
1 d
f (A) − f (B) −
f (B + αX)
∈ I1 ,
k!
dα
α=0
j=1
(2.7)
and there is a distribution T with
Z
k
n−1
X
1 d
Tr f (A) − f (B) −
f (B + αX)
=
T (λ)f (n) (λ) dλ.
k! dα
R
α=0
j=1
(2.8)
Moreover, the distribution T is such that Tb ∈ L∞ (R; dt).
Proof. This is immediate from the estimates in Proposition 2.1 and
Z
−1/2
f (A) = (2π)
fb(t) eitA dt.
R
For, by (2.5), we have
kLHS of (2.7)kI1 ≤ C
Z
|t| |fb(t)| dt = C
n
R
Thus, (2.8) defines a distribution T with
Z
b dt,
|T (f )| ≤ C |f(t)|
Z
R
(n) (t)| dt.
|fd
(2.9)
R
so Tb is a function in L∞ (R; dt).
Notice that, as we have seen,
Z t
d it(B+αX) e
=i
eiβB Xei(t−β)B dβ
dα
α=0
0
ON THE KOPLIENKO SPECTRAL SHIFT FUNCTION, I. BASICS
so that formally,
7
d
′
Tr
f (B + αX)
− Xf (B) = 0,
dα
α=0
and formally, (1.5) can replace the right-hand side of (1.4). This can
be proven if the commutator [B, X] = [B, A] is trace class.
Dostanić [19, Theorem 2.9] essentially proves that T is the derivative
of an L2 (R; dλ)-function.
A key point for us is that in the case n = 2, the distribution T
is given by an L1 (R; dλ)-function. We start this construction here by
considering the trace class case.
Lemma 2.3. Let B be a self-adjoint operator and let X = (A − B) ∈
I1 . Then there is a (complex ) measure dµB,X on R such that for any
bounded Borel function, f ,
Z
Tr(Xf (B)) =
f (λ) dµB,X (λ).
(2.10)
R
Equation (2.10) defines dµB,X uniquely.
Proof. Equation (2.10) yields uniqueness of the measure dµB,X since
it defines the integral for all continuous functions f . Regarding existence of dµB,X , the spectral theorem asserts the existence of measures
dµB;ϕ,ψ , such that
Z
hϕ, f (B)ψi =
f (λ) dµB;ϕ,ψ (λ)
(2.11)
R
and
kµB;ϕ,ψ k ≤ kϕk kψk.
(2.12)
The canonical decomposition for X (see [60, Sect. 1.2]) says (with N
finite or infinite)
N
X
X=
µj (X)hϕj , · iψj ,
(2.13)
j=1
where
{ψj }N
j=1
and
{ϕj }N
j=1
are orthonormal sets, µj > 0, and
N
X
µj (X) = kXkI1 .
(2.14)
j=1
Define
dµB,X =
N
X
µj (X) dµB;ϕj ,ψj
(2.15)
j=1
which converges by (2.12) and (2.14).
8
F. GESZTESY, A. PUSHNITSKI, AND B. SIMON
Theorem 2.4. Let A, B be bounded operators and X = (A − B) ∈ I1 .
Let ξ(λ; A, B) be the KrSSF. Let dµB,X be given by (2.10). Define
Z λ
η(λ; A, B) ≡ µB,X ((−∞, λ)) −
ξ(λ′ ; A, B) dλ′, λ ∈ R. (2.16)
−∞
Then η( · ; A, B) has compact support and for any f ∈ C ∞ (R), we have
Z
d
Tr f (A) − f (B) −
f (B + αX)
=
f ′′ (λ)η(λ; A, B) dλ.
dα
R
α=0
(2.17)
Remarks. 1. Since
Z
Z
dµB,X (λ) = Tr(X) =
ξ(λ; A, B) dλ,
R
R
we can replace (−∞, λ) in both places in (2.16) by [λ, ∞). This shows
that η in (2.16) has compact support.
2. (2.17) determines η uniquely up to an affine term. The condition
that η have compact support (as the η of (2.16) does) determines η
uniquely.
d
Proof. We first claim that dα
f (B + αX)α=0 is trace class and that
d
Tr
f (B + αX)
= Tr(Xf ′ (B)).
(2.18)
dα
α=0
This is immediate for f nice enough (e.g., f such that (2.6) holds for
n = 1) since
Tr(eiαβB Xeiα(1−β)B ) = Tr(XeiαB ).
(2.19)
Thus, by (2.10),
Z
d
Tr
f (B + αX) =
f ′ (λ) dµB,X (λ)
dα
R
Z
= − f ′′ (λ)[µB,X ((−∞, λ))].
R
Similarly, by (1.6),
Z
f ′ (λ)ξ(λ; A, B) dλ
R
Z λ
Z
′′
′
′
= − f (λ)
ξ(λ ; A, B) dλ dλ.
Tr(f (A) − f (B)) =
R
−∞
The next critical step will be to prove positivity of η.
ON THE KOPLIENKO SPECTRAL SHIFT FUNCTION, I. BASICS
9
3. Convexity of Tr(f (A))
Positivity of η is essentially equivalent to the following result:
Theorem 3.1. Let f be a convex function on R. Then the mapping
A 7→ Tr(f (A))
(3.1)
is a convex function on the m×m self-adjoint matrices for every m ∈ N.
Remarks. 1. More generally, if f is convex on (a, b), (3.1) is convex
on matrices A with spectrum in (a, b). In fact, it is easy to see that
any convex function f on (a, b) is a monotone limit on (a, b) of convex functions on R. So this more general result is a consequence of
Theorem 3.1.
2. We will discuss the infinite-dimensional situation below.
Two special cases of this are widely known and used:
(a) A 7→ Tr(eA ) is convex.
(b) A 7→ Tr(A log(A)) is convex on A ≥ 0.
Both of these are rather special. In the first case, one has the stronger
A 7→ log(Tr(eA )) is convex and the usual proof of it is via Hölder’s
inequality (cf., e.g., [28, p. 19–20] or [58, p. 57]) which proves the strong
convexity of the log(·), but does not prove Theorem 3.1. In the second
case, by Kraus’ theorem [37, 8] A 7→ A log(A) is operator convex. (We
also note that Ar , r ∈ R, is operator convex for A > 0 if and only if
r ∈ [−1, 0] ∪ [1, 2] (cf. [9, p. 147]).) We have found Theorem 3.1 stated
in Alicki–Fannes [3, Sect. 9.1] and an equivalent statement in Ruelle
[55, Sect. 2.5] (who attributes it to Klein [35] although Klein only has
the special case f (x) = x log(x) and his proof is specific to that case;
Ruelle’s is not). We have also found it in Lieb–Pedersen [45] whose
proof is closer to the one we label “Third Proof” below. The result
is also mentioned in von Neumann [66], although the proof he gives
earlier for a special f does not seem to establish the general case.
In any event, even though this result is not hard and is known to some
experts, we provide several proofs because it is not widely known and
is central to the theory of KoSSF. We provide several proofs because
they illustrate different aspects of the theorem.
First Proof. This uses eigenvalue perturbation theory. By a limiting
argument, it suffices to prove it for functions f ∈ C ∞ (R). By approximating derivatives of f by polynomials, we see that matrix elements,
and so the trace of f (A), are C ∞ -functions of A. By a limiting argument, we need only show λ → Tr(A + λX) has a nonnegative second
derivative at λ = 0 in case A has distinct eigenvalues.
10
F. GESZTESY, A. PUSHNITSKI, AND B. SIMON
So by changing basis, we suppose A is diagonal with the eigenvalues
a1 < a2 < · · · < am . Let ej (λ) be the eigenvalue of A + λX near aj for
|λ| sufficiently small. As is well known [33, Sect. II.2], [53, Sect. XII.1],
m
2
X
|Xk,j
|
d2 ej =
.
(3.2)
dλ2 λ=0
aj − ak
k=1
k6=j
Clearly,
d
[f (ej (λ))] = f ′ (ej (λ))e′j (λ),
dλ
d2
f (ej (λ)) = f ′′ (ej (λ))e′j (λ)2 + f ′ (ej (λ))e′′j (λ),
dλ2
d2
Tr(f
(A(λ)))
= 1 + 2 ,
dλ2
λ=0
so
where
1 =
m
X
(3.3)
f ′′ (aj )e′j (0)2 ≥ 0
j=1
′′
since f ≥ 0 and, by (3.2),
2 =
m
X
|Xk,j |
j,k=1
k6=j
2
f ′ (aj ) − f ′ (ak )
≥0
aj − ak
since for x < y,
f ′ (y) − f ′ (x)
1
=
y−x
y−x
Z
y
f ′′ (u) du ≥ 0.
x
Second Proof. This one uses a variational principle. We consider first
the case
(
x, x ≥ 0,
f + (x) = x+ ≡
(3.4)
0, x < 0.
We claim first that
Tr(f + (A)) = max{Tr(AB) | kBk ≤ 1, B ≥ 0},
(3.5)
where k · k is the matrix norm on Cm with the Euclidean norm. For in
an orthonormal basis where A is a diagonal matrix,
Tr(AB) =
m
X
j=1
aj bj,j ≤
m
X
j=1
(aj )+ = Tr(f + (A))
ON THE KOPLIENKO SPECTRAL SHIFT FUNCTION, I. BASICS
11
if 0 ≤ bjj ≤ 1. On the other hand, if B is the diagonal matrix with
(
1, aj > 0,
bjj =
0, aj ≤ 0,
then B ≥ 0, kBk ≤ 1, and Tr(AB) = Tr(f + (A)). This proves (3.5).
Convexity is immediate for f + given by (3.4) once we have (3.5), since
maxima of linear functionals are convex. Obviously, since (x − λ)+ is
just a translate of x+ , we get convexity for any function of
Z ∞
(x − λ)+ dµ(λ)
λ0
for any Borel measure µ on (λ0 , ∞). But every convex function f with
f ≡ 0 for x ≤ λ0 has this form.
Adding ax + b to this, we get the result for any convex function f
with f ′′ (x) = 0 for x ≤ λ0 . Taking λ0 → −∞, we get the result for
general convex functions f .
Third Proof (M. B. Ruskai, private communication). If f is any convex function, C a self-adjoint m × m matrix with
Cej = λj ej ,
(3.6)
and v ∈ Cm a unit vector, then
hv, f (C)vi =
m
X
|hv, ej i|2f (λj )
j=1
≥f
X
m
λj |hv, ej i|
j=1
2
= f (hv, Cvi),
where (3.7) employs Jensen’s inequality.
Now suppose
C = θA + (1 − θ)B,
θ ∈ [0, 1].
Then,
Tr(f (C)) =
=
m
X
j=1
m
X
j=1
f (hej , Cej i)
f (θhej , Aej i + (1 − θ)hej , Bej i)
(3.7)
(3.8)
12
F. GESZTESY, A. PUSHNITSKI, AND B. SIMON
≤
m
X
[θf (hej , Aej i) + (1 − θ)f (hej , Bej i)]
j=1
m
X
≤θ
hej , f (A)ej i + (1 − θ)
j=1
m
X
hej , f (B)ej i
(3.9)
(3.10)
j=1
= θTr(f (A)) + (1 − θ)Tr(f (B)),
(3.11)
proving convexity. In the above, (3.9) is direct convexity of f and
(3.10) is (3.8) for v = ej and C = A or B.
Corollary 3.2. If f ∈ C 1 (R) is convex and B and X are m × m
self-adjoint matrices, m ∈ N, then
d
Tr f (B + X) − f (B) −
f (B + αX)
≥ 0.
(3.12)
dα
α=0
Remarks. 1. It is not hard to see that (3.12) is equivalent to Theorem 3.1.
2. It is in this form that the result appears in Ruelle [55, Sect. 2.5],
and for the case f (x) = x log(x), x > 0, in Klein [35].
Proof. If g ∈ C 1 (R) is a convex function,
g(x + y) − g(x) − g ′(x)y ≥ 0,
(3.13)
since convexity says that g lies above the tangent line at any point.
(3.12) is (3.13) for g(α) = Tr(f (B + αX)), x = 0, y = 1.
Corollary 3.3. For finite-dimensional matrices A and B, the KoSSF,
η( · ; A, B), satisfies η(λ; A, B) ≥ 0 for a.e. λ ∈ R.
Proof. Let h : R → [0, ∞) be a measurable function bounded and
supported on an interval (a, b) with σ(A) ∪ σ(B) ⊂ (a, b) (so, by (2.16),
η is supported on (a, b)). Let f be the unique convex function with
f = 0 near −∞ and f ′′ = h. By (3.12) and (2.17),
Z
0≤
h(λ)η(λ; A, B) dλ.
(3.14)
R
Since h is arbitrary, η ≥ 0 a.e.
Theorem 3.4. For any finite self-adjoint matrices A, B (of the same
size),
Z
|η(λ; A, B)| dλ = 12 kA − Bk2I2 .
(3.15)
R
ON THE KOPLIENKO SPECTRAL SHIFT FUNCTION, I. BASICS
13
Remarks. 1. It is remarkable that we always have equality in (3.15).
The analog for the KrSSF is
Z
|ξ(λ; A, B)| dλ ≤ kA − BkI1 ,
(3.16)
R
where equality, in general, holds if A − B is either positive or negative.
2. (3.15) emphasizes again the lack of a chain rule for η; η is nonlinear
in (A − B).
Proof. Take f (x) = 12 x2 such that f ′′ (x) = 1 and
d
f (B + X) − f (B) −
f (B + αX)
dα
α=0
2
1
= 2 (B + X) − B 2 − XB − BX = 12 X 2 .
R
R
Since η ≥ 0, R f ′′ (λ)η(λ; A, B) dλ = R |η(λ; A, B)| dλ and (3.15) holds.
In Section 5 we take limits from the finite-dimensional situation, but
one can easily extend Theorem 3.1 in two ways and from there directly
prove η ≥ 0 and (3.15) in case (A − B) ∈ I1 . Without proof, we state
the extensions (the results are simple limiting arguments from finite
dimensions):
Theorem 3.5. If f is convex on R and f (0) = 0, then f (A) is trace
class for any self-adjoint trace class operator A, and for such A’s, the
mapping A 7→ Tr(f (A)) is convex.
In this context we note that convex functions are Lipschitz continuous. (For this and additional regularity results of convex functions,
see, e.g., [9, p. 145–146].)
Theorem 3.6. For any convex function f ∈ C ∞ (R), any bounded selfadjoint operator B, and any self-adjoint operator X ∈ I1 ,
[f (B + X) − f (B)] ∈ I1
and the mapping X 7→ Tr(f (B + X) − f (B)) is convex.
Convexity of maps of the type s 7→ Tr(f (B + X(s)) − f (B)), s ∈
(s1 , s2 ), for convex f and certain classes of X(·) ∈ I1 was also studied
in [24].
4. Löwner’s Formula and the Finite-Dimensional
Birman–Solomyak Bound
The final element needed to construct the KoSSF is the following
lovely theorem of Birman–Solomyak [13] (see also [15]):
14
F. GESZTESY, A. PUSHNITSKI, AND B. SIMON
Theorem 4.1. Let A, B be bounded self-adjoint operators with (A−B)
Hilbert–Schmidt. Let f be a function defined on an interval [a, b] ⊃
σ(A) ∪ σ(B). Suppose f is uniformly Lipschitz, that is,
kf kL = sup
x,y∈[a,b]
x6=y
|f (x) − f (y)|
< ∞.
|x − y|
(4.1)
Then [f (A) − f (B)] is also Hilbert–Schmidt and
kf (A) − f (B)kI2 ≤ kf kL kA − BkI2 .
(4.2)
The proof in [13] depends on the deep machinery of double Stieltjes
operator integrals. Our two points in this section are:
(1) The inequality for finite matrices is quite elementary and, by limits, extends to (4.2).
(2) The key to our proof, a kind of “Double Stieltjes Operator Integral for Dummies,” goes back to Löwner [46] in 1934 whose contributions to this theme seem not to have been appreciated in the
literature on double Stieltjes operator integrals.
Given two finite m × m self-adjoint matrices A, B with respective
m
m
m
eigenvectors {ϕj }m
j=1 and {ψj }j=1 and eigenvalues {xj }j=1 and {yj }j=1
such that
Aϕj = xj ϕj ,
Bψj = yj ψj ,
(4.3)
we introduce the (modified) Löwner matrix of a function f by
(
f (yk )−f (xℓ )
, yk 6= xℓ ,
yk −xℓ
Lk,ℓ =
1 ≤ k, ℓ ≤ m.
0,
yk = xℓ ,
(4.4)
(Löwner [46] originally supposed yk 6= xℓ for all 1 ≤ k, ℓ ≤ m.) Clearly,
if f is Lipschitz,
sup |Lk,ℓ | ≤ kf kL.
(4.5)
1≤k,ℓ≤m
Löwner noted that since
f (A)ϕj = f (xj )ϕj ,
f (B)ψj = f (yj )ψj ,
(4.6)
we have Löwner’s formula:
hψk , [f (B) − f (A)]ϕℓ i = Lkℓ hψk , (B − A)ϕℓ i,
(4.7)
and this holds even if yk = xℓ (since then both matrix elements vanish).
This is the “baby” version of the double Stieltjes operator integral
formula
Z
Z
f (y) − f (x)
f (B) − f (A) =
dEB (x)(B − A)dEA (y)
y−x
σ(A) σ(B)
ON THE KOPLIENKO SPECTRAL SHIFT FUNCTION, I. BASICS
15
due to Birman and Solomyak [13, 15]. Here the integration is with
respect to the spectral measures of A and B.
Löwner’s formula immediately implies:
Proposition 4.2. (4.2) holds for finite self-adjoint matrices.
Proof. Hilbert–Schmidt norms can be computed in any basis, even two
different ones, that is,
kCk2I2 =
m
X
kCϕℓ k2 =
ℓ=1
m
X
|hψk , Cϕℓ i|2 .
k,ℓ=1
Thus,
kf (A) − f (B)k2I2 =
=
m
X
k,ℓ=1
m
X
|hψk , (f (B) − f (A))ϕℓ i|2
L2kℓ |hψk , (B − A)ϕℓ i|2
by (4.7)
k,ℓ=1
≤
kf k2L
m
X
|hψk , (B − A)ϕℓ i|2
by (4.5)
k,ℓ=1
= kf k2LkA − Bk2I2 .
Proof of Theorem 4.1. Let {ζj }∞
j=1 be an orthonormal basis for H and
PN the orthogonal projections onto the linear span of {ζj }N
j=1 . For any
A and B and Lipschitz f , by Proposition 4.2,
kf (PN BPN ) − f (PN APN )kI2 ≤ kf kLkPN (B − A)PN kI2
≤ kf kLkB − AkI2
(4.8)
(4.9)
if B − A is Hilbert–Schmidt, since
kPN (B −
A)PN k2I2
=
n
X
2
kPN (B − A)ζj k ≤
j=1
n
X
k(B − A)ζj k2
j=1
(4.10)
Ak2I2 .
(4.11)
k[f (PN BPN ) − f (PN APN )]ζj k2 ≤ kf k2LkB − Ak2I2 .
(4.12)
≤ kB −
Thus, for any k ∈ N,
k
X
j=1
16
F. GESZTESY, A. PUSHNITSKI, AND B. SIMON
As PN BPN −→ B strongly, one infers by continuity of the funcN →∞
tional calculus that f (PN BPN ) −→ f (B) strongly. Since the sum in
N →∞
(4.12) is finite, one concludes that
k
X
k(f (B) − f (A))ζj k2 ≤ kf k2L kB − Ak2I2 .
j=1
Taking k → ∞, we see that [f (B)−f (A)] ∈ I2 and that (4.2) holds. 5. General Construction of the KoSSF η( · ; A, B)
The general construction and proof of properties of η depends first
on an approximation of trace class operators by finite rank ones and
then on an approximation of Hilbert–Schmidt operators by trace class
operators. In this section, we mostly follow the approach of [36, Lemma
3.3].
Theorem 5.1. Let Bn , B, n ∈ N, be uniformly bounded self-adjoint operators such that Bn −→ B strongly. Let Xn , X, n ∈ N, be a sequence
n→∞
of self-adjoint trace class operators such that kX − Xn kI1 −→ 0. Then
n→∞
for any continuous function, g, of compact support, we conclude that
Z
Z
g(λ)η(λ; Bn + Xn , Bn ) dλ −→
g(λ)η(λ; B + X, B) dλ. (5.1)
n→∞
R
R
In particular, η( · ; A, B) ≥ 0 a.e. on R if (A − B) ∈ I1 and, in that
case, (3.15) holds.
Proof. By Theorem A.7 and
Z λ
Z
Z
′
′
g(λ)
ξ(λ ; A, B) dλ dλ =
R
−∞
∞
′
ξ(λ ; A, B)
−∞
Z
∞
g(λ) dλ dλ′
λ′
we get convergence of the second term in (2.16). By (2.10),
dµBn ,Xn −→ dµB,X
n→∞
weakly by the strong continuity of the functional calculus since
|Tr(Xn f (Bn )) − Tr(Xf (B))|
≤ |Tr(X[f (Bn ) − f (B)])| + kf k∞ kX − Xn kI1 −→ 0,
n→∞
as f is continuous.
Since weak limits of positive measures are positive, the positivity follows from positivity in the finite-dimensional case taking Bn = Pn BPn
and Xn = Pn XPn for finite-dimensional Pn converging strongly to I,
the identity operator.
ON THE KOPLIENKO SPECTRAL SHIFT FUNCTION, I. BASICS
17
Once we have positivity, we obtain (3.15) directly by following the
proof of Theorem 3.4.
Theorem 5.2. Let A, B, C be bounded self-adjoint operators such that
(A − C) ∈ I1 and (B − C) ∈ I1 . Then
Z
|η(λ; A, C) − η(λ; B, C)| dλ ≤ kA − BkI2 12 kA − BkI2 + kB − CkI2 .
R
(5.2)
Proof. We begin with (1.9) which follows from the fact that (2.17)
holds when (A − B) ∈ I1 . Here (1.10) holds for nice functions g, say,
g ∈ C ∞ (R). Thus,
Z
Z
LHS of (5.2) ≤ |η(λ; A, B)| dλ + |δη(λ; A, B, C)| dλ.
(5.3)
R
R
By (3.15),
First term on RHS of (5.3) ≤ 12 kA − BkI2 kA − BkI2 .
(5.4)
As for δη, by (1.10),
Z
g ′ (λ)δη(λ) dλ ≤ kA − BkI2 kg(B) − g(C)kI2
R
≤ kg ′k∞ kA − BkI2 kB − CkI2
by Theorem 4.1. Since δη ∈ L1 (R) and the bounded C ∞ (R)-functions
are k·k∞ -dense in the bounded continuous functions, and for h ∈ L1 (R),
Z
khk1 = sup f (x)h(x) dx,
f ∈C(R)
kf k∞ =1
R
we conclude
kδηk1 ≤ kA − BkI2 kB − CkI2 .
Relations (5.3)–(5.5) imply (5.2).
(5.5)
Here is the main theorem on the existence of the KoSSF:
Theorem 5.3. Let A, B be two bounded self-adjoint operators with
(B −A) ∈ I2 . Then there exists a unique L1 (R; dλ)-function η( · ; A, B)
supported on (− max(kAk, kBk), max(kAk, kBk)) such that for any g ∈
C ∞ (R),
d
g(B + α(A − B))
∈ I1
(5.6)
g(A) − g(B) −
dα
α=0
18
F. GESZTESY, A. PUSHNITSKI, AND B. SIMON
and
Z
d
Tr g(A) − g(B) −
g(B + α(A − B))
=
g ′′ (λ)η(λ; A, B) dλ.
dα
R
α=0
(5.7)
Moreover,
Z
R
η( · ; A, B) ≥ 0 a.e. on R,
|η(λ; A, B)| dλ = 21 kA − Bk2I2 ,
(5.8)
(5.9)
and for any bounded self-adjoint operators A, B, C with (A − C) ∈ I2
and (B − C) ∈ I2 ,
Z
|η(λ; A, C) − η(λ; B, C)| dλ ≤ kA − BkI2 12 kA − BkI2 + kB − CkI2 .
R
(5.10)
Remark. For a sharp condition on the class of functions η for which
Koplienko’s trace formula holds, we refer to Peller [52].
Proof. Let X = A − B and pick Xn ∈ I1 , n ∈ N, such that kXn −
XkI2 −→ 0. By (5.10), which we have proven for
n→∞
η(λ; B + Xn , B) − η(λ; B + Xm , B),
we see that η( · ; B + Xn , B) is Cauchy in L1 (R; dλ) and so converges
a.e. to what we will define as η( · ; A, B). (5.7) holds by taking limits;
η ≥ 0 as a limit of positive functions η. (5.9) and (5.10) hold by taking
limits.
Remark. Here is an alternative method of proving the estimate (5.2),
bypassing Theorem 5.1: In the same way as in Corollary 3.3, one can
deduce positivity of η( · ; A, B) for (A−B) ∈ I1 from Theorem 3.6. The
only place where Theorem 5.1 is used in the proof of Theorem 5.2 is
in the estimate (5.4). This estimate (see Theorem 3.4) follows directly
from the positivity of η and the trace formula (2.15).
6. What Functions η are Possible?
We introduce the classes of functions
η(I2 ) = {η( · ; A, B) | A, B bounded and self-adjoint, (A − B) ∈ I2 },
η(I1 ) = {η( · ; A, B) | A, B bounded and self-adjoint, (A − B) ∈ I1 }.
In this section we would like to raise the question of the description of
the classes η(I2 ) and η(I1 ).
ON THE KOPLIENKO SPECTRAL SHIFT FUNCTION, I. BASICS
19
Since, for now, we are considering A, B bounded, η has compact
support.
First we discuss the class η(I2 ). By Theorem 5.3, all functions of this
class are nonnegative and Lebesgue integrable. It would be interesting
to see if the class η(I2 ) contains all nonnegative Lebesgue integrable
functions.
As a step towards answering this question, we give the following
elementary result:
Theorem 6.1. The class η(I2 ) contains all nonnegative Riemann integrable functions of compact support.
Remark. A contemporary account of the theory of Riemann integrable
functions can be found, for instance, in Stein–Shakarchi [62].
Proof. First we consider a simple example. Let a ∈ R and ε > 0;
consider the operators in C2 given by the diagonal 2 × 2 matrices B =
diag(a − ε, a + ε) and A = diag(a + ε, a − ε). Then the KoSSF for this
pair is given by (cf. (2.16)) η(λ) = 2εχ(a−ǫ,a+ǫ) (λ). We note that
Z
η(λ) dλ = 4ε2 = 12 Tr((A − B)2 ),
R
in agreement with (5.9).
Next, suppose that 0 ≤ η ∈ L1 (R; dλ) is represented by the
L1 (R; dλ)-convergent series
η(λ) =
∞
X
|In |χIn (λ),
(6.1)
n=1
where In ⊂ R are (not necessarily disjoint) finite intervals and |In | is
the length of In . Denote by an the midpoint of In and let εn = 12 |In |.
We introduce
∞
B = ⊕∞
n=1 diag(an − εn , an + εn ) and A = ⊕n=1 diag(an + εn , an − εn )
2
1
in the Hilbert space ⊕∞
(R; dλ)-convergence of
n=1 C . Note that the L P
∞
2
the series (6.1) is equivalent to the condition
n=1 εn < ∞ and so
A − B = ⊕∞
n=1 diag(2εn , −2εn ) is a Hilbert–Schmidt operator. It is
clear that the KoSSF for the pair A, B coincides with η.
Thus, it suffices to prove that any Riemann integrable function 0 ≤
η ∈ L1 (R; dλ) can be represented as an L1 (R; dλ)-convergent series
(6.1).
Let 0 ≤ η ∈ L1 (R; dλ) be Riemann integrable. According to the
definition of the Riemann integral, there exists a finite set of disjoint
20
F. GESZTESY, A. PUSHNITSKI, AND B. SIMON
open squares Qn , n ∈ {1, . . . , M}, which fit under the graph of η and
M
X
area(Qn ) ≥
1
2
n=1
Z
η(λ) dλ.
R
In other words, there exists a finite set of (not necessarily disjoint) open
intervals In ⊂ R, n ∈ {1, . . . , N}, such that
N
X
|In |χIn (λ) ≤ η(λ),
n=1
Z X
N
R
n=1
λ ∈ R,
Z
N
X
2
1
|In |χIn (λ) dλ =
|In | ≥ 2
η(λ) dλ.
R
n=1
Thus, we can represent η as
η(λ) =
N1
X
|In |χIn (λ) + η1 (λ),
n=1
Z
η1 (λ) ≥ 0,
η1 (λ) dλ ≤
R
1
2
Z
η(λ) dλ,
R
and the sum is taken over a finite set of indices n ∈ {1, . . . , N1 }. Iterating this procedure, we see that for any m ∈ N we can represent η
as
η(λ) =
Nm
X
n=1
ηm (λ) ≥ 0,
|In |χIn (λ) + ηm (λ),
Z
ηm (λ) dλ ≤ 2
−m
R
Z
η(λ) dλ,
R
where εn > 0, an ∈ R, and the sum is taken over a finite set of indices n.
Taking m → ∞, it follows that η can be represented as an L1 (R; dλ)convergent series (6.1).
Regarding the class η(I1 ), we note only that every function of this
class is of bounded variation. This follows from (2.16), since both
terms on the right-hand side of (2.16) are of bounded variation. We
also note that it follows from the proof of Theorem 6.1 that the class
η(I1 ) contains all functions of the type
η(λ) =
∞
X
n=1
|In |χIn (λ),
∞
X
n=1
|In | < ∞.
ON THE KOPLIENKO SPECTRAL SHIFT FUNCTION, I. BASICS
21
7. Modified Determinants and the KoSSF
In this section, as a preliminary to the next, we want to use our
viewpoint to prove a formula for modified perturbation determinants
in terms of the KoSSF originally derived by Koplienko [36]. We recall
that one of Krein’s motivating formulas for the KrSSF is (see (A.32)):
Z
−1
−1
det((A − z)(B − z) ) = exp
(λ − z) ξ(λ) dλ , z ∈ C\R. (7.1)
R
Here det(·) is the Fredholm determinant defined on I+I1 (since A−B =
X ∈ I1 implies (A − z)(B − z)−1 − I = X(B − z)−1 ∈ I1 ); see [27, 60].
We recall that for C ∈ I1 , one can define det2 (·) by
det2 (I + C) = det(I + C)e−Tr(C)
(7.2)
and that C 7→ det2 (I + C) extends uniquely and continuously to I2 ,
the Hilbert–Schmidt operators, although the right-hand side of (7.2)
no longer makes sense (see [27, Ch. IV], [60, Ch. 9]). Our goal in this
section is to prove the following formula first derived by Koplienko [36]:
Theorem 7.1. Let B and X be bounded self-adjoint operators and X ∈
I2 . Let A = B + X. Then for any z ∈ C\R, (A − z)(B − z)−1 ∈ I + I2
and
Z
η(λ; A, B)
−1
det2 ((A − z)(B − z) ) = exp −
dλ .
(7.3)
2
R (λ − z)
Proof. It suffices to prove (7.3) for X ∈ I1 since both sides are continuous in I2 norm and I1 is dense in I2 . Continuity of the left-hand
side follows from Theorem 9.2(c) of [60] and of the right-hand side by
Theorem 5.2 above.
When X ∈ I1 , we can use (7.2). Let
Z λ
g1 (λ) = µB,X ((−∞, λ)), g2 (λ) =
ξ(λ′ ; A, B) dλ′.
(7.4)
−∞
By an integration by parts argument (using g2′ = ξ),
Z
Z
ξ(λ)
g2 (λ)
dλ =
dλ.
2
R λ−z
R (λ − z)
(7.5)
By an integration by parts in a Stieltjes integral and by (2.10),
Z
1
−1
Tr(X(B − z) ) =
dµB,X (λ)
(7.6)
R λ−z
Z
1
dλ.
(7.7)
=
g1 (λ)
(λ − z)2
R
22
F. GESZTESY, A. PUSHNITSKI, AND B. SIMON
Thus, by (7.1) and (7.2),
Z
−2
det2 (1 + X(B − z) ) = exp − (g1 (λ) − g2 (λ))(λ − z) dλ ,
−1
R
which, given (2.16) is (7.3).
8. On Boundary Values of Modified Perturbation
Determinants det2 ((A − z)(B − z)−1 )
−1
By (7.1), if (A − B) ∈ I1 , det((A
R − z)(B−1− z) ) has a limit as
z → λ + i0 for a.e. λ ∈ R since R (ν − z) ξ(ν) dν is a difference
of Herglotz functions. In this section, we will consider nontangential
boundary values to the real axis of modified perturbation determinants
det2 ((A − z)(B − z)−1 ),
z ∈ C+ ,
where X = (A − B) ∈ I2 . Unlike the trace class, we will see nontangential boundary values may not exist a.e. on R.
For notational simplicity in the remainder of this section, we now
abbreviate KoSSF simply by η, that is, η ≡ η( · ; A, B).
In contrast to the usual (trace class) SSF theory, we have the following nonexistence result for boundary values of modified perturbation
determinants:
Theorem 8.1. There exists a pair of self-adjoint operators A, B (in
a complex, separable Hilbert space) such that X = (A − B) ∈ I2 ,
σ(B) is an interval, and for a.e. λ ∈ σ(B), the nontangential limit
limz→λ, z∈C+ det2 (I + X(B − zI)−1 ) does not exist.
Proof. By Theorems 6.1 and 7.1, the proof reduces to the following
statement: There exists a Riemann integrable 0 ≤ η ∈ L1 (R; dλ) with
support being an interval such that for a.e. λ ∈ supp (η), the nontangential limit
Z
η(λ) dλ
(8.1)
lim
z→λ R (λ − z)2
z∈C+
does not exist.
First we note that the existence of the limit in (8.1) at the point λ
depends only on the behavior of η(t) when t varies in a small neighborhood of λ. Thus, it suffices to construct 0 ≤ η ∈ L1 (R; dλ) such
that the limits (8.1) do not exist for a.e. λ ∈ (−1, 1); by shifting and
scaling such a function η, one obtains the required statement for a.e.
λ ∈ σ(B).
Let us first obtain the required example of η defined on the unit
circle ∂D, and then transplant it onto the real line. By a well-known
ON THE KOPLIENKO SPECTRAL SHIFT FUNCTION, I. BASICS
23
construction employing either lacunary series or Rademacher functions
(see [21], [22, App. A], [70, I, p.
there exists a power series f (z) =
P
P6]),
∞
∞
n
n=1 cn z , |z| ≤ 1, such that
n=1 |cn | < ∞ and for a.e. z ∈ ∂D, the
limit limζ→z f ′ (ζ) does not exist as ζ approaches z from inside of the
unit disc along any nontangential trajectory. By construction, Im(f )
is continuous on ∂D and
Z
Z
Im(f (ζ))
1
Im(f (ζ))
1
′
f (z) =
dζ, f (z) =
dζ, |z| < 1.
π ∂D (ζ − z)
π ∂D (ζ − z)2
Let a > − minζ∈∂D Im(f (ζ)) and set v(ζ) = Im(f (ζ))+a if |arg ζ| < π/2
and v(ζ) = 0 otherwise. Then v ≥ 0 and v is piecewise continuous (with
the possible discontinuities only for arg(ζ) = ±π/2); in particular,
v is Riemann integrable. Again by a localization argument, for a.e.
θ ∈ (−π/2, π/2), the limit
Z
Im(f (ζ))
lim
dζ
2
iθ
z→e
∂D (ζ − z)
does not exist as z approaches eiθ from inside of the unit disc along
any nontangential trajectory.
It remains to transplant v from the unit circle onto the real line. Let
t = i 1−ζ
, w = i 1−z
, and η(t) = v(ζ(t)). Then 0 ≤ η ∈ L1 (R; dλ),
1+ζ
1+z
supp (η) ⊂ (−1, 1), η is Riemann integrable, and
Z
Z
Im(f (ζ))
(w + i)2 1 η(λ) dλ
dζ = −
.
2
2
2i
−1 (λ − w)
∂D (ζ − z)
Thus, the limit (8.1) does not exist for a.e. λ ∈ (−1, 1).
9. KoSSF for Unbounded Operators
In this section we briefly discuss the question of existence of KoSSF
under the assumption
[(A − z)−1 − (B − z)−1 ] ∈ I2
(9.1)
instead of (A − B) ∈ I2 . This question was studied in [49] and [52] (see
also [36] for related issues).
First recall the invariance principle for the KrSSF. Assume that A, B
are bounded self-adjoint operators and (A − B) ∈ I1 . Let ϕ = ϕ ∈
C ∞ (R), ϕ′ 6= 0 on R. Then we have
ξ(λ; A, B) = sign (ϕ′ ) ξ(ϕ(λ); ϕ(A), ϕ(B)) + const for a.e. λ ∈ R.
(9.2)
This is a consequence of Krein’s trace formula (1.6). With an appropriate choice of normalization of KrSSF, the constant in the right-hand
side of (9.2) vanishes.
24
F. GESZTESY, A. PUSHNITSKI, AND B. SIMON
When both (A − B) ∈ I1 and [ϕ(A) − ϕ(B)] ∈ I1 , formula (9.2) is an
easily verifiable identity. But when [ϕ(A)−ϕ(B)] ∈ I1 yet (A−B) 6∈ I1 ,
this formula can be regarded as a definition of ξ( · ; A, B).
In contrast to this, no explicit formula relating η(ϕ(·); ϕ(A), ϕ(B)) to
η( · ; A, B) is known. The reason is simple: The definition of η involves
not only a trace formula but a choice of interpolation A(θ) between
B and A. For bounded self-adjoint operators, the choice A(θ) = (1 −
θ)B + θA, θ ∈ [0, 1], is natural. But when one only has (9.1), what
choice does one make? It is natural to define A(θ) by
(A(θ) − z)−1 = (1 − θ)(B − z)−1 + θ(A − z)−1 ,
θ ∈ [0, 1].
(9.3)
For this to be self-adjoint, we need z ∈ R, which means we should have
some real point in the intersection of the resolvent sets for A and B.
Even if there were such a z, it is not unique and the interpolation will
not be unique. Moreover, the convexity that led to η ≥ 0 may be lost.
The net result is that the situation, both after the work of others and
our work, is less than totally satisfactory.
Let us discuss a certain surrogate of (9.2) for the KoSSF. The formulas below are a slight variation on the theme of the construction of
[49].
First assume that A and B are bounded operators and X = (A−B) ∈
I2 . Let δ ⊂ R be an interval which contains the spectra of A and B
and ϕ ∈ C ∞ (δ), ϕ′ 6= 0. Denote a = ϕ(A), b = ϕ(B), x = a−b. By the
Birman–Solomyak bound (1.7), we have x ∈ I2 and so both η( · ; A, B)
and η( · ; a, b) are well defined. Let us display the corresponding trace
formulas:
Z
d
f (B + αX)
=
η(λ; A, B)f ′′(λ) dλ,
Tr f (A) − f (B) −
dα
R
α=0
(9.4)
Z
d
Tr g(a) − g(b) −
g(b + αx)
=
η(µ; a, b)g ′′(µ) dµ.
dα
R
α=0
(9.5)
Now suppose f = g ◦ ϕ. In contrast to the corresponding calculation
for the KrSSF, the left-hand sides of (9.4) and (9.5) are, in general,
distinct. However, we can make the right-hand sides look similar if we
introduce the following modified KoSSF:
′
Z λ
1
1
ηe(λ; A, B) = η(ϕ(λ); a, b) ′
−
η(ϕ(t); a, b)
dt. (9.6)
ϕ (λ)
ϕ′ (t)
λ0
The choice of λ0 above is arbitrary; it affects only the constant term in
the definition of ηe.
ON THE KOPLIENKO SPECTRAL SHIFT FUNCTION, I. BASICS
25
By a simple calculation involving integration by parts, we get
Z
Z
′′
η(µ; a, b)g (µ) dµ =
ηe(λ; A, B)f ′′ (λ) dλ, f = g ◦ ϕ ∈ C0∞ (R).
R
R
(9.7)
Combining (9.5) and (9.7), we get the modified trace formula
Z
d
−1
Tr f (A) − f (B) −
f ◦ ϕ (b + αx)
=
ηe(λ; A, B)f ′′(λ) dλ
dα
R
α=0
(9.8)
∞
for all f ∈ C0 (R). Precisely as for the KrSSF, one can treat (9.6) and
(9.8) as the definition of a modified KoSSF ηe( · ; A, B).
We consider an example of this construction which might be useful in
applications. Suppose that A and B are lower semibounded self-adjoint
operators such that for some (and thus for all) z ∈ C\(σ(A) ∪ σ(B))
the inclusion (9.1) holds. Choose E ∈ R such that inf σ(A + E) > 0
1
and inf σ(B + E) > 0. Take ϕ(λ) = λ+E
and let a = (A + E)−1 ,
b = (B + E)−1 , x = a − b. For λ > −E, define
ηe(λ; A, B) = −η((λ + E)−1 ; a, b)(λ + E)2
Z λ
+2
η((t + E)−1 ; a, b)(t + E) dt.
(9.9)
−E
Note that η( · ; a, b) is integrable and η(λ; a, b) vanishes for large λ and
therefore the integral in (9.9) converges. Moreover, this definition ensures that ηe(λ; A, B) = 0 for λ < inf(σ(A) ∪ σ(B)). Thus, it is natural
to define
ηe(λ; A, B) = 0 for λ ≤ −E.
(9.10)
The above calculations prove the following result:
Theorem 9.1. Let A, B, a, b, x be as above. Then there exists a
function ηe( · ; A, B) such that
Z
ηe(λ; A, B)(λ + E)−4 dλ < ∞
(9.11)
R
and ηe(λ; A, B) = 0 for λ < inf(σ(A) ∪ σ(B)) and for all f ∈ C0∞ (R)
the following trace formula holds:
Z
d
−1
Tr f (A) − f (B) − f ((b + αx) − E)
=
ηe(λ; A, B)f ′′(λ) dλ.
dα
R
α=0
(9.12)
We note that condition (9.11) does not fix the linear term in the
definition of ηe but (9.10) does.
26
F. GESZTESY, A. PUSHNITSKI, AND B. SIMON
In [49], a pair of self-adjoint operators A, B was considered under the
assumption (9.1) alone (without the lower semiboundedness assumption). Another regularization of η( · ; A, B) was suggested in this case.
The construction of [49] is more intricate than the above calculation
and uses KoSSF for unitary operators.
In [36], the assumption
(A − B)|A − iI|−1/2 ∈ I2
was used. This assumption is intermediate between (A − B) ∈ I2 and
(9.1). Under this assumption, the trace formula (5.7) was proven with
0 ≤ η ∈ L1 (R; (1 + λ)−γ dλ) for any γ > 12 .
Finally, in [49], the assumption
(A − B)(A − iI)−1 ∈ I2
was used and formula (5.7) was proven with η ∈ L1 (R; (1 + λ2 )−2 dλ).
Note that the difference between the last two results and Theorem 9.1
is that in Theorem 9.1, a modified trace formula (9.12) is proven rather
than the original formula (5.7). Theorem 9.1 is nothing but a change
of variables in the trace formula for resolvents, whereas the abovementioned results of [36] and [49] require some work.
10. The Case of Unitary Operators
In this section, we want to briefly discuss a definition of η for a pair
of unitaries. Once again, there is an issue of interpolation. If A and B
are the unitaries,
A(θ) = (1 − θ)B + θA,
θ ∈ [0, 1],
(10.1)
∞
is not unitary, so we cannot define f (A(θ)) for arbitrary C -functions
on ∂D = {z ∈ C | |z| = 1}. Neidhardt [49] (see also [52]) discussed one
way of interpolating by writing A = eC , B = eD for suitable C and D
and interpolating, but there is considerable ambiguity in how to choose
C, D as well as whether to look at eθC+(1−θ)D or e(1−θ)D eθC , etc.
Here, with Szegő’s theorem as background [25], we want to discuss
an alternative to Neidhardt’s approach.
Lemma 10.1. Let A, B be unitary with (A − B) ∈ I2 . Then for any
n = 0, 1, 2, . . . ,
d
n
n
n
A(θ) ∈ I1 .
(10.2)
A −B −
dθ
θ=0
We have
n
d
n
A − Bn −
≤ n(n − 1) kA − Bk2I .
A(θ)
(10.3)
2
dθ
2
θ=0 I1
ON THE KOPLIENKO SPECTRAL SHIFT FUNCTION, I. BASICS
27
In fact,
LHS of (10.3) = o(n2 ).
(10.4)
Proof. Let X = A − B. Then, by telescoping,
n
n
A(θ) − B =
n−1
X
A(θ)j (θX)B n−1−j .
(10.5)
j=0
Thus, since kA(θ)k ≤ 1, kBk = 1,
kA(θ)n − B n kI2 ≤ n|θ| kXkI2
(10.6)
kA(θ)n − B n k ≤ 2.
(10.7)
and, of course,
Dividing (10.5) by θ and taking θ to zero yields
n−1
so
X
d
A(θ)n =
B j X B n−1−j ,
dθ
j=0
LHS of (10.2) =
n−1
X
(Aj − B j )X B n−1−j .
(10.8)
j=0
(10.3) is immediate since (10.8) and (10.6) implies
X
n−1 2
LHS of (10.3) ≤ kXk2
j .
(10.9)
j=0
(1)
(2)
To get (10.4), we write X = Xε + Xε
(1)
kXε kI1 < ∞. Thus, (10.8) implies
(2)
where kXε kI2 ≤ ε and
n(n − 1)
+ 2nkXε(1) kI1
2
using (10.7) instead of (10.6). Dividing by n2 , taking n → ∞, and
then ε ↓ 0, show
LHS of (10.3) ≤ εkXkI1
lim sup n−2 LHS of (10.3) = 0.
n→∞
Theorem 10.2. Let A and B be unitary so (A − B) ∈ I2 . Then there
exists a real distribution η(λ; A, B)
on ∂D so that for any polynomial
d
P (z), P (A) − P (B) − dθ P (A(θ)) θ=0 ∈ I2 and
Z 2π
d
dθ
Tr P (A) − P (B) −
P (A(θ))
=
P ′′ (eiθ )η(eiθ ; A, B) .
dθ
2π
0
θ=0
(10.10)
28
F. GESZTESY, A. PUSHNITSKI, AND B. SIMON
Moreover, the moments of η satisfy
Z 2π
dθ
einθ η(eiθ )
= o(1).
(10.11)
2π |n|→∞
0
R 2π
dθ
Remarks. 1. As usual, we use 0 f (eiθ )η(eiθ ) 2π
as shorthand for the
distribution η acting on the function f .
2. As we will discuss, η is determined by (10.10) up to three real
constants in an affine term.
3. For a sharp condition on the class of functions for which Neidhardt’s version of Koplienko’s trace formula for unitary operators holds,
we refer to Peller [52].
Proof. Let cn , n ∈ Z, be defined by


0,
cn = [n(n − 1)]−1 Tr An − B n −

c ,
−n
d
dθ
A(θ)n θ=0
n = 0, 1,
(10.12)
, n ≥ 2,
n ≤ −1.
By Lemma 10.1, cn = o(1) as n → ∞, so there is a distribution η =
η( · ; A, B) satisfying
Z 2π
dθ
cn =
ei(n−2)θ η(eiθ ) , n ≥ 2.
(10.13)
2π
0
By (10.12), we have (10.10) for P (z) = z n for n ≥ 2 and both
sides are zero for P (z) = z m , m = 0, 1. Thus, (10.10) holds for all
polynomials.
For any c0 ∈ R, c1 ∈ C, we can add c0 + c1 eiθ + c̄1 e−iθ to η without
changing the right-hand side of (10.10). We wonder if η is always in
L1 (∂D) with η ≥ 0 for some choice of c0 and c1 . The condition cn → 0
is, of course, consistent with η ∈ L1 (∂D).
11. Open Problems and Conjectures
While we have found some new aspects of η here and summarized
much of the prior literature, there are many open issues. The most
important one concerns properties of η and the invariance of the a.c.
spectrum:
Conjecture 11.1. Suppose A, B are self-adjoint with (A − B) ∈ I2
and that on some interval (a, b) ⊂ σ(A) ∩ σ(B), we have η( · ; A, B)
and η( · ; B, A) are of bounded variation with distributional derivatives
in Lp ((a, b); dλ) on (a, b) for some p > 1. Then σac (A) ∩ (a, b) =
σac (B) ∩ (a, b).
ON THE KOPLIENKO SPECTRAL SHIFT FUNCTION, I. BASICS
29
In the appendix, we prove the invariance for I1 -perturbations using
boundary values of det((A − z)(B − z)−1 ). When η has the properties
in the conjecture, det2 ((A − z)(B − z)−1 ) has boundary values and we
hope those can be used to get the invariance of a.c. spectrum. While
we made the conjecture assuming control of η( · ; A, B) and η( · ; B, A),
we wonder if only one suffices. Similarly, we wonder if Lp , p > 1, can
be replaced by the weaker condition that the derivative is a sum of an
L1 -piece and the Hilbert transform of an L1 -piece.
Open Question 11.2. Is the η we constructed in Section 10 for the
unitary case an L1 (∂D) function?
Open Question 11.3. Is the class η(I2 ) introduced in Section 6 all of
L1 (R; dλ) (of compact support ), or only the Riemann integrable functions, or something in between?
Open Question 11.4. Is the class η(I1 ) all functions of bounded variation or a subset, and if so, what subset?
Appendix: On the KrSSF ξ( · ; A, B)
Both for comparison and because the Krein spectral shift (KrSSF)
is needed in our construction of the KoSSF, we present the basics of
the KrSSF here. Most of the results in this appendix are known (see,
e.g., [7, Sect. 19.1.4]), [14], [16], [38], [39], [40], [61], [64], [68, Ch. 8],
[69] and the references therein) so this appendix is largely pedagogical,
but our argument proving the invariance of a.c. spectrum under trace
class perturbations at the end of this appendix is new. Moreover, we
fill in the details of an approach sketched in [60, Ch. 11] exploiting the
method Gesztesy–Simon [26] used to construct the rank-one KrSSF.
Most approaches define ξ via perturbation determinants.
We will need the following strengthening of Theorem 2.2:
Theorem A.1. Let f be a function of compact support whose Fourier
transform fb satisfies (2.6) for n = 1 (in particular, f can be C 2+ε (R)).
Then,
(a) For any bounded self-adjoint operators A, B with (A − B) ∈ I1 ,
(f (A) − f (B)) ∈ I1 . Moreover,
where
b 1 kA − BkI ,
kf (A) − f (B)kI1 ≤ kk fk
1
kk fbk1 ≤
Z
R
k|fb(k)| dk.
(A.1)
(A.2)
30
F. GESZTESY, A. PUSHNITSKI, AND B. SIMON
(b) Let Bn , B, n ∈ N, be uniformly bounded self-adjoint operators such
that Bn −→ B strongly. Let Xn , X, n ∈ N, be a sequence of selfn→∞
adjoint trace class operators such that kX − Xn kI1 −→ 0. Then,
n→∞
Tr(f (Bn + Xn ) − f (Bn )) −→ Tr(f (B + X) − f (B)).
n→∞
(A.3)
Proof. (a) is immediate from Proposition 2.1 which implies
f (A) − f (B)
= (2π)
−1/2
Z
R
ik fb(k)
Z
1
iβkA
e
i(1−β)kB
(A − B)e
0
dβ dk.
(A.4)
This also implies (b) via the dominated convergence theorem, continuity of the functional calculus (so Cn −→ C strongly implies
n→∞
eitCn −→ eitC strongly), and the fact that if Xn −→ X in I1
n→∞
n→∞
and Cn −→ C strongly (with Cn , C uniformly bounded), then
n→∞
Tr(Cn Xn ) −→ Tr(CX). This latter fact comes from
n→∞
|Tr(Cn Xn − CX)| ≤ |Tr(Cn (Xn − X) − (C − Cn )X)|
≤ kCn k kXn − X1 kI1 + |Tr((C − Cn )X)|
P
µm (X)hϕm , · iψm , then
X
|Tr((Cn − C)X)| ≤
µm (X)|hϕm , (Cn − C)ψm i| −→ 0
and if X =
m∈N
n→∞
m∈N
by the dominated convergence theorem.
Part (a) in Theorem A.1, in a slightly more general form, is stated
and proved in [40, p. 141].
Now let B be a bounded self-adjoint operator and ϕ a unit vector.
For α ∈ R, define
Aα = B + α(ϕ, · )ϕ
(A.5)
and for z ∈ C\R,
Fα (z) = (ϕ, (Aα − z)−1 ϕ),
(A.6)
Gα (z) = 1 + αF0 (z).
(A.7)
The resolvent formula implies (see [60, Sect. 11.2])
Fα (z) =
F0 (z)
,
1 + αF0 (z)
z ∈ C\R,
(A.8)
ON THE KOPLIENKO SPECTRAL SHIFT FUNCTION, I. BASICS
31
and that
(Aα − z)−1 − (B − z)−1
α
(A.9)
=−
((B − z̄)−1 ϕ, · )(B − z)−1 ϕ, z ∈ C\R,
1 + αF0 (z)
implying
α
Tr((B − z)−1 − (Aα − z)−1 ) =
(ϕ, (B − z)−2 ϕ)
1 + αF0 (z)
(A.10)
d
=
log(Gα (z)), z ∈ C\R.
dz
Theorem A.2. Let B be a bounded self-adjoint operator and Aα given
by (A.5) for α ∈ R and ϕ with kϕk = 1. Then for a.e. λ ∈ R,
1
ξα (λ) = lim arg(Gα (λ + iε))
(A.11)
π ε↓0
exists and satisfies
(i)
0 ≤ ±ξα ( · ) ≤ 1 if 0 < ±α.
(A.12)
(ii) ξα (λ) = 0 if λ ≤ min(σ(Aα ) ∪ σ(B)) or λ ≥ max(σ(Aα ) ∪ σ(B)).
(iii)
Z
|ξα (λ)| dλ = |α|
(A.13)
(iv) For any z ∈ C\R,
Gα (z) = exp
Z
R
(v)
(λ − z) ξα (λ) dλ .
−1
det((Aα − z)(B − z)−1 ) = Gα (z).
(vi) For any z ∈ C\R,
Z
−1
−1
Tr((B − z) − (Aα − z) ) = (λ − z)−2 ξα (λ) dλ.
(A.14)
(A.15)
(A.16)
R
(vii) For any f satisfying the hypotheses of Theorem A.1,
Z
Tr(f (Aα ) − f (B)) =
f ′ (λ)ξα(λ) dλ.
(A.17)
R
Remarks. 1. This theorem and its proof are essentially the same as the
starting point of Krein’s construction in [38] (see also [40, p. 134–136]
or [16, Sect. 3]).
2. In (A.11), arg(Gα (z)) is defined uniquely for Im(z) > 0 by demanding continuity in z and
lim arg(Gα (iy)) = 0.
y↑∞
(A.18)
32
F. GESZTESY, A. PUSHNITSKI, AND B. SIMON
For Im(z) < 0 one has Gα (z) = Gα (z).
3. By (A.9), (Aα − z)(B − z)−1 is of the form I+ rank one, and so
lies in I + I1 . The det(·) in (A.15) is the Fredholm determinant (see
[60, Ch. 3]). This is the same as the finite-dimensional determinant
det(C) for I + D with D finite rank and C = (I + D) ↾ K where K is
any finite-dimensional space containing ran(D) and (ker(D))⊥ .
4. The exponential Herglotz representation basic to this proof goes
back to Aronszajn and Donoghue [5].
5. Comparing (A.17) and (1.6), one concludes
ξα ( · ) = ξ( · ; A, B).
Proof. By the spectral theorem, there is a probability measure dµα (λ)
such that
Z
dµα (λ)
Fα (z) =
.
(A.19)
R λ−z
In particular,
Im(F0 (z)) > 0 if Im(z) > 0,
(A.20)
so on C+ = {z ∈ C | Im(z) > 0},
± Im(Gα (z)) > 0 if ± α > 0.
(A.21)
Since Gα (iy) → 1, as y ↑ ∞, we can define
log(Gα (z)) = Hα (z)
on C+ uniquely if we require Hα (iy) −→ 0. By (A.21),
y↑∞
0 < ± Im(Hα ( · )) ≤ π.
(A.22)
By the general theory of Herglotz functions (see, e.g., [4], [5]), the
limit in (A.11) exists and (A.12) holds by (A.22). (A.22) also implies
that the limiting measure w-limε↓0 ± π1 Im(Hα (λ+iε)) dλ in the Herglotz
representation theorem is purely absolutely continuous, hence (A.14)
holds.
(A.16) then follows from (A.14) and (A.10).
Since
Fα (z) = −z −1 + O(z −2 ),
(A.23)
z→∞
(A.17) implies
Gα (z) = 1 − αz −1 + O(z −2 ),
z→∞
and thus (A.14) implies
Z
R
ξα (λ) dλ = α,
(A.24)
(A.25)
ON THE KOPLIENKO SPECTRAL SHIFT FUNCTION, I. BASICS
33
which, given (A.12), implies (A.13). This proves everything except the
parts (ii), (v), and (vii).
To prove (A.15), we note that with Pϕ = (ϕ, · )ϕ, we have
(Aα − z)(B − z)−1 = I + αPϕ (B − z)−1 ,
which, since Pϕ is rank one, implies
det((Aα − z)(B − z)−1 ) = 1 + Tr(αPϕ (B − z))
= 1 + αF0 (z)
= Gα (z).
Let us prove (ii) for α > 0. The proof of α < 0 is similar. Let
a = min(σ(B)), b = max(σ(B)). Then, by (A.19),
Z
dµ0 (λ)
′
>0
F0 (x) =
2
R (λ − x)
on (−∞, a)∪(b, ∞) and F0 −→ 0. Thus, F > 0 on (−∞, a) and F < 0
x→±∞
on (b, ∞). Let f = limx↓b F (x) which may be −∞. If 1 + αf < 0, there
is a unique c with 1+αF0(c) = 0, and then Gα is positive on (c, ∞). By
(A.8), Fα (z) is analytic away from (a, b)∪{c}. Thus, σ(Aα ) ∈ (a, b)∪{c}
and c = max(σ(Aα ), σ(B)), so (ii) says that ξα (λ) = 0 on (−∞, b) and
(c, ∞). Since Gα (x) > 0 there and 0 < arg(Gα (z + iε)) < π, we see
that ξα (x) = 0 on these intervals.
Finally, we turn to (vii). Since B n − Anα can be written as a telescoping series, it is trace class and
kB n − Anα kI1 ≤ n [sup(kAα k, kBk)]n−1kB − AkI1 .
(A.26)
Thus, both sides of (A.16) are analytic about z = ∞, so identifying
Taylor coefficients,
Z
n
n
Tr(B − Aα ) =
nλn−1 ξα (λ) dλ.
(A.27)
R
Summing Taylor series for ezλ , using (A.26) and (A.27) proves (ezB −
ezAα ) ∈ I1 and
Z
zB
zAα
(A.28)
Tr(e − e ) = z ezλ ξα (λ) dλ.
R
This leads to (A.17) by using (A.4).
In extending this, the following uniqueness result will be useful:
34
F. GESZTESY, A. PUSHNITSKI, AND B. SIMON
Proposition A.3. Suppose A and B are bounded self-adjoint operators
and (A − B) ∈ I1 . Suppose ξj ∈ L1 (R; dλ) for j = 1, 2, and for all
f ∈ C0∞ (R),
Z
Tr(f (A) − f (B)) =
f ′ (λ)ξj (λ) dλ.
(A.29)
R
Then ξ1 = ξ2 . Moreover, if (a, b) ⊂ R\σ(A) ∪ σ(B), ξj (·) is an integer
on (a, b), and if a = −∞ or b = ∞, it is zero on (a, b), and so ξj has
compact support.
Proof. By (A.29), the distribution ξ1 − ξ2 has vanishing distributional
derivative, so is constant. Since it lies in L1 (R; dλ), it must be zero.
If f ∈ C0∞ ((a, b)), f (A) = f (B) = 0, so ξj′ has zero derivative on (a, b)
and so is constant. If a = −∞ or b = ∞, the constant must be zero
since ξj ∈ L1 (R; dλ). Now pick f which is supported on (c, (a+b)/2) for
some c < d < min(σ(A) ∪ σ(B)) with f = 1 on (d, (3a + b)/4). Thus,
the right-hand side of (A.29) is the negative of the constant value of ξj
on (a, b), while the left-hand side is the trace of a trace class difference
of projections which is always an integer (see [6, 23]).
Theorem A.4. For any pair of bounded self-adjoint operators A, B
with (A − B) of finite rank, there exists a function, ξ( · ; A, B) such
that the following hold:
(i) (A.17) holds for any f satisfying the hypotheses of Theorem A.1.
(ii)
|ξ( · ; A, B)| ≤ rank(A − B).
(A.30)
(iii)
Z
|ξ(λ; A, B)| dλ ≤ kA − BkI1 .
(A.31)
R
(iv) For z ∈ C\R, one has
−1
det((A − z)(B − z) ) = exp
Z
(λ − z) ξ(λ) dλ .
−1
R
(A.32)
(v) ξ(λ) = 0 for λ ≤ min(σ(A) ∪ σ(B)) or λ ≥ max(σ(A) ∪ σ(B)).
(vi) If (A − B) and (B − C) are both finite rank,
ξ( · ; A, C) = ξ( · ; A, B) + ξ( · ; B, C).
(A.33)
Proof. If (A − B) has rank n, we can find A0 = A, A1 , . . . , An = B so
(Aj+1 − Aj ) has rank one, and
n−1
X
j=0
kAj+1 − Aj kI1 = kB − AkI1 .
(A.34)
ON THE KOPLIENKO SPECTRAL SHIFT FUNCTION, I. BASICS
35
We define
ξ( · ; A, B) =
n−1
X
ξ( · ; Aj , Aj+1 ),
(A.35)
j=0
where ξ( · ; Aj , Aj+1) is constructed via Theorem A.2. (A.17) holds
by telescoping and the rank-one case. (A.30) and (A.31) follow from
(A.12), (A.13), and (A.34).
(A.32) follows from
(A − z)(B − z)−1 = [(A0 − z)(A1 − z)−1 ][(A1 − z)(A2 − z)−1 ] . . .
using
det((1 + X1 )(1 + X2 )) = det(1 + X1 ) det(1 + X2 )
for X1 , X2 ∈ I1 .
Item (v) is proven in Proposition A.3. Item (vi) follows from the
uniqueness in Proposition A.3.
Theorem A.4 is essentially the same as Theorem 3 in [38] (see also
[40] and [16]).
Corollary A.5. If A, A′ are both finite rank perturbations of B with
all three operators self-adjoint, we have
Z
|ξ(λ; A, B) − ξ(λ; A′, B)| dλ ≤ kA − A′ kI1 .
(A.36)
R
Proof. By (A.33),
ξ( · ; A, B) − ξ( · ; A′, B) = ξ( · ; A, A′).
Thus, (A.36) follows from (A.31).
This yields the principal result on existence and properties of the
KrSSF (see [38] or [40]).
Theorem A.6. Let A, B be bounded self-adjoint operators with (A −
B) ∈ I1 . Then,
(i) There exists a unique function ξ( · ; A, B) ∈ L1 (R; dλ) such that
(A.17) holds for any f satisfying the hypotheses of Theorem A.1.
(ii)
Z
|ξ(λ; A, B)| dλ ≤ kA − BkI1 .
(A.37)
R
(iii)
(iv)
(v)
(vi)
(A.32) holds.
ξ(λ) = 0 if λ ≤ min(σ(A) ∪ σ(B)) or λ ≥ max(σ(A) ∪ σ(B)).
If (A − B) and (B − C) are both trace class, (A.33) holds.
If (A − B) and (A′ − B) are trace class, (A.36) holds.
36
F. GESZTESY, A. PUSHNITSKI, AND B. SIMON
Proof. Find An so (An − B) −→ (A − B) in I1 and (An − B) is finite
n→∞
rank. By (A.36), ξ( · ; An, B) is Cauchy in L1 (R) so converges to an
L1 (R) function by (A.36). Thus, items (i), (ii), (iii), (v), and (vi) hold
by taking limits (using k·kI1 -continuity of the mapping C → det(I +C).
Uniqueness and (iv) follow from Proposition A.3.
We refer to [50] (see also [51]) for a description of a class of functions
f for which this theorem holds.
We note that there are interesting extensions of the trace formula
(A.17) to classes of operators A, B different from self-adjoint or unitary
operators. While we cannot possibly list all such extensions here, we
refer, for instance, to Adamjan and Neidhardt [1], Adamjan and Pavlov
[2], Jonas [29], [30], Krein [41], Langer [44], Neidhardt [47], [48], Rybkin
[56], Sakhnovich [57], and the literature cited therein.
Theorem A.7. Let Bn , B, n ∈ N, be uniformly bounded self-adjoint
operators such that Bn −→ B strongly. Let Xn , X, n ∈ N, be a sen→∞
quence of self-adjoint trace class operators such that kX −Xn kI1 −→ 0.
n→∞
Then for any continuous function, g,
Z
Z
g(λ)ξ(λ; Bn + Xn , Bn ) dλ −→
g(λ)ξ(λ; B + X, B) dλ. (A.38)
R
n→∞
R
Proof. By Theorem A.1, we have (A.38) for g ∈ C0∞ (R). Note
that the k · k∞ -norm closure of C0∞ includes the continuous functions of compact support. Thus, by an approximation argument
using uniform L1 (R; dλ)-bounds on ξ, we get (A.38) for continuous functions of compact support. Since ξ(λ; A, B) = 0 for λ ∈
[− max(kAk, kBk), max(kAk, kBk)], the result for continuous functions
of compact support extends to any continuous function.
We want to note the following. Define
ξ(I1 ) = {ξ( · ; A, B) | A, B bounded and self-adjoint, (A − B) ∈ I1 }.
Proposition A.8. ξ(I1 ) is the set of L1 (R; dλ)-elements of compact
support.
Proof. Since A, B are bounded and self-adjoint, any ξ( · ; A, B) ∈ ξ(I1 )
necessarily lies in L1 (R; dλ) and has compact support (cf. Theorem
A.6 (i) and (iv)).
Next, let g ∈ L1 (R; dλ) satisfy 0 ≤ g(λ) ≤ 1 and supp(g) ⊂ (a, b) for
some −∞ < a < b < ∞. Define
Z b
1
g(λ) dλ
G(z) = exp
, Im(z) > 0.
(A.39)
π a λ−z
ON THE KOPLIENKO SPECTRAL SHIFT FUNCTION, I. BASICS
37
Then G satisfies the following items (i)–(iii):
(i) Im(G(z)) > 0 (for Im(z) > 0) since
Z b
Z b
g(λ) dλ
dλ
0 ≤ Im
≤ Im
≤π
λ−z
a
a λ−z
on account of 0 ≤ g ≤ 1.
(ii) Im(G(λ + i0)) = 0 if λ < a or λ > b.
(iii) G(z) → 1 as Im(z) → ∞ since g ∈ L1 (R; dλ). It follows that there
is α > 0 and a probability measure dµ on [a, b] with
Z b
dµ(λ)
G(z) = 1 + α
.
(A.40)
a λ−z
Let B be multiplication by λ on L2 ((a, b); dµ), ϕ is the function 1 in
L2 ((a, b); dµ) and A = B + α(ϕ, · )ϕ. Then, by (A.5), (A.6), (A.7), and
Rb
(A.14), ξ(λ; A, B) = π −1 g(λ) for a.e. λ ∈ (a, b), and α = π −1 a g(λ) dλ.
Thus, we have the theorem if 0 ≤ g ≤ 1 or (by interchanging A and
B) if 0 ≥ g ≥ −1. Since any L1 (R; dλ)-function is a sum of such g’s
converging in L1 (R; dλ) (simple functions are dense in L1 (R; dλ)), we
obtain the general result.
We note that a similar result for the finite rank case can be found in
[42].
Finally, we prove invariance of the absolutely continuous spectrum
under trace class perturbations using the KrSSF and perturbation determinants, that is, without directly relying on elements from scattering
theory.
We start with the following observations:
Lemma A.9. Let A, B be bounded self-adjoint operators with X =
(A − B) of rank one. Then,
σac (A) = σac (B)
and
ξ(λ; A, B) ∈ {−1, 0, 1}
for a.e. λ ∈ R\σac (B).
Proof. ξ(λ; A, B) ∈ {−1, 0, 1} follows from (A.5)–(A.7), (A.11), and
(A.12). σac (A) = σac (B) follows in the usual manner by computing the
normal boundary values to the real axis of the imaginary part of Fα in
terms of that of F0 using (A.8).
Lemma A.10. Let A, B be bounded self-adjoint operators with X =
(A − B) ∈ I1 . Then for a.e. λ ∈ R\σac (B) one has
lim det(I + X(B − λ − iε)−1 ) ∈ R.
ε↓0
38
F. GESZTESY, A. PUSHNITSKI, AND B. SIMON
Proof. By (A.32), it suffices to prove that
ξ( · ; A, B) ∈ Z a.e. on R\σac (B).
(A.41)
Introducing
X=
∞
X
xn (φn , · )φn,
X0 = 0, XN =
n=1
N
X
xn (φn , · )φn , N ∈ N,
n=1
the rank-by-rank construction of ξ( · ; A, B) alluded to in the proof of
Theorem A.6 yields the L1 (R; dλ)-convergent series
∞
X
ξ( · ; A, B) =
ξ( · ; B + Xn , B + Xn−1 ).
(A.42)
n=1
By Lemma A.9, each term in the above series is integer-valued a.e. on
R\σac (B) and hence so is the left-hand side of (A.42), which yields
(A.41).
Lemma A.11. Let A, B be bounded self-adjoint operators in the Hilbert
space H with X = (A − B) ∈ I1 and ϕ ∈ H, kϕk = 1. Denote
Pϕ = (ϕ, · )ϕ. Then,
1 − (ϕ, (B − z)−1 ϕ) =
det(I − (X + Pϕ )(A − z)−1 )
,
det(I − X(A − z)−1 )
z ∈ C+ . (A.43)
Proof. One computes
I − Pϕ (B − z)−1 = (B − Pϕ − z)(A − z)−1 (A − z)(B − z)−1
= (B − Pϕ − z)(A − z)−1 [(B − z)(A − z)−1 ]−1
= [I − (X + Pϕ )(A − z)−1 ][I − X(A − z)−1 ]−1 .
(A.44)
Taking determinants in (A.44) then yields
det(I − (X + Pϕ )(A − z)−1 )
= det(I − Pϕ (B − z)−1 )
det(I − X(A − z)−1 )
= 1 − (ϕ, (B − z)−1 ϕ).
Theorem A.12. Let A, B be bounded self-adjoint operators in the
Hilbert space H with (A − B) ∈ I1 . Then,
σac (A) = σac (B).
Proof. By symmetry between A and B, it suffices to prove σac (B) ⊆
σac (A). Suppose to the contrary that there exists a set E ⊆ σac (B)
such that |E| > 0 and E ∩ σac (A) = ∅. Choose an element ϕ ∈ H such
that limε↓0 Im((ϕ, (B − λ − iε)−1 )ϕ) > 0 for a.e. λ ∈ E. Thus, for a.e.
λ ∈ E, the imaginary part of the limit z → λ + i0 of the left-hand
ON THE KOPLIENKO SPECTRAL SHIFT FUNCTION, I. BASICS
39
side of (A.43) is nonzero. On the other hand, by Lemma A.10, the
right-hand side of (A.43) is real for a.e. λ ∈ E, a contradiction.
Remark. Employing det(I − A) = det2 (I − A)eTr(A) for A ∈ I1 , and
using an approximation of Hilbert–Schmidt operators by trace class
operators in the norm k · kI2 , one rewrites (A.43) in the case where
X = (A − B) ∈ I2 as
1 − (ϕ, (B − z)−1 ϕ) =
det2 (I − (X + Pϕ )(A − z)−1 ) (ϕ,(A−z)−1 ϕ)
e
,
det2 (I − X(A − z)−1 )
z ∈ C+ . (A.45)
Since in the proof of Theorem A.12 one assumes E ⊆ σac (B), |E| > 0,
and E ∩ σac (A) = ∅, one concludes that
(ϕ, (A − λ − i0)−1 ϕ) is real-valued for a.e. λ ∈ E.
Moreover, if the boundary values of det2 (I − X(A − λ − i0)−1 ) exist for
a.e. λ ∈ σac (B), by (A.45), so do those of det2 (I − (X + Pϕ )(A − λ −
i0)−1 ). Hence, if one can assert real-valuedness of
det2 (I − (X + Pϕ )(A − λ − i0)−1 )
for a.e. λ ∈ σac (B),
(A.46)
det2 (I − X(A − λ − i0)−1 )
using input from some other sources, one can follow the proof of Theorem A.12 step by step to obtain invariance of the a.c. spectrum.
In the special case of Schrödinger (and similarly for Jacobi) operators
with real-valued potentials V ∈ Lp ([0, ∞)), p ∈ [1, 2], the existence of
the boundary values of det2 (I − X(A − λ − i0)−1 ) is indeed known due
to Christ–Kiselev [17] (for p ∈ [1, 2) using some heavy machinery) and
Killip–Simon [34] (for p = 2). We will return to this circle of ideas in
[25].
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